4 - Orders in simple Artin rings

Summary

In this chapter we study rings which are right orders in simple Artin rings and explicitly determine these via the Goldie and Faith–Utumi Theorems. We also present some further results concerning representations of semiprime maximal orders. Although the rings considered are more general than simple Goldie rings, the conclusions and the techniques of proof are invaluable in the structure theory of simple Goldie rings.

Recall that a right ideal I of R is called nil if for all x ∈ I, xⁿ = 0 for some n > 0.

The identity (ra)^{n + 1} = r(ar)ⁿa for elements a, r ∈ R shows that Ra is a nil left ideal iff aR is a nil right ideal. Thus, R has no nil right ideals ≠ 0 if and only if R has no nil left ideals ≠ 0.

4.1 LemmaLet S be a ring satisfying the maximum condition for right annulets. If S has a nonzero nil right or left ideal A, then S contains a nonzero nilpotent ideal.

Proof (Utumi) By the preceding remark, we can assume A is a nil left ideal ≠ 0. Let 0 ≠ a ∈ A be such that a^⊥ is maximal in {x^⊥ ∣ 0 ≠ x ∈ A}. If u ∈ S is such that ua ≠ 0, then (ua)ⁿ = 0 and (ua)^{n− 1} ≠ 0 for some n > 1.